The bounds are sharp for \(p=q\) , since otherwise they would imply sharper bounds in the Besov scale than are known to be possible, by (REF ). Moreover, the following radial Knapp example shows sharpness of the polynomial growth bound for general \(p,q\) , since the \(\mathcal {B}^{s}_{p,q,q}(\mathbb {R}^{d})\)  norm coincides with the \(L^{p}(\mathbb {R}^{d})\)  norm for low frequencies. The analog for the Schrödinger equation was considered in [1]}.
